Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#11498 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C2×C22⋊C8 |
| prime radical | C23×C79 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 127 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,20,25,22,27,35,43,23,28,36,44,21,26,34,42)(19,29,30,46,31,40,47,41,32,45,33,48,37,24,38,39)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,24)(20,33)(21,37)(22,31)(23,32)(25,41)(26,45)(27,29)(28,40)(30,36)(34,47)(35,38)(39,44)(42,46)(43,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,22,35,23,36,21,34)(18,25,27,43,28,44,26,42)(19,30,31,47,32,33,37,38)(24,39,29,46,40,41,45,48)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 4486);